Properties

Label 9.5.d.a
Level 9
Weight 5
Character orbit 9.d
Analytic conductor 0.930
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 9.d (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.930329667755\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.39400128.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{2} \) \( + ( -3 + \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{3} \) \( + ( 6 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4} \) \( + ( -3 - \beta_{1} - 3 \beta_{5} ) q^{5} \) \( + ( -14 - 7 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} + \beta_{4} + \beta_{5} ) q^{6} \) \( + ( 4 + 14 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{7} \) \( + ( -30 + \beta_{1} + \beta_{2} - 54 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{8} \) \( + ( 45 - 6 \beta_{1} + 9 \beta_{2} + 45 \beta_{3} + 3 \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{2} \) \( + ( -3 + \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{3} \) \( + ( 6 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4} \) \( + ( -3 - \beta_{1} - 3 \beta_{5} ) q^{5} \) \( + ( -14 - 7 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} + \beta_{4} + \beta_{5} ) q^{6} \) \( + ( 4 + 14 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{7} \) \( + ( -30 + \beta_{1} + \beta_{2} - 54 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{8} \) \( + ( 45 - 6 \beta_{1} + 9 \beta_{2} + 45 \beta_{3} + 3 \beta_{4} ) q^{9} \) \( + ( 2 - 8 \beta_{1} + 8 \beta_{2} - \beta_{4} - \beta_{5} ) q^{10} \) \( + ( 108 - 2 \beta_{2} + 54 \beta_{3} - 3 \beta_{4} ) q^{11} \) \( + ( -8 + 21 \beta_{1} + \beta_{2} - 114 \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{12} \) \( + ( -25 - 13 \beta_{1} - 26 \beta_{2} - 20 \beta_{3} + 10 \beta_{4} - 5 \beta_{5} ) q^{13} \) \( + ( -114 + 2 \beta_{1} + 135 \beta_{3} + 21 \beta_{5} ) q^{14} \) \( + ( -175 + 13 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} ) q^{15} \) \( + ( -30 - 42 \beta_{1} - 21 \beta_{2} - 41 \beta_{3} + 15 \beta_{4} - 30 \beta_{5} ) q^{16} \) \( + ( -60 - 3 \beta_{1} - 3 \beta_{2} - 162 \beta_{3} - 21 \beta_{4} + 21 \beta_{5} ) q^{17} \) \( + ( 336 + 33 \beta_{1} + 3 \beta_{2} + 207 \beta_{3} - 24 \beta_{4} + 3 \beta_{5} ) q^{18} \) \( + ( -52 + 9 \beta_{1} - 9 \beta_{2} + 9 \beta_{4} + 9 \beta_{5} ) q^{19} \) \( + ( 324 - 28 \beta_{2} + 162 \beta_{3} + 24 \beta_{4} ) q^{20} \) \( + ( -56 - 74 \beta_{1} + 19 \beta_{2} - 342 \beta_{3} + 15 \beta_{4} - 8 \beta_{5} ) q^{21} \) \( + ( -62 + 65 \beta_{1} + 130 \beta_{2} - 60 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{22} \) \( + ( -69 + 29 \beta_{1} + 27 \beta_{3} - 42 \beta_{5} ) q^{23} \) \( + ( -82 + 16 \beta_{1} + 65 \beta_{2} + 219 \beta_{3} - 13 \beta_{4} + 38 \beta_{5} ) q^{24} \) \( + ( 70 + 2 \beta_{1} + \beta_{2} + 127 \beta_{3} - 35 \beta_{4} + 70 \beta_{5} ) q^{25} \) \( + ( -282 - 26 \beta_{1} - 26 \beta_{2} - 486 \beta_{3} + 39 \beta_{4} - 39 \beta_{5} ) q^{26} \) \( + ( 27 - 72 \beta_{1} - 162 \beta_{2} + 216 \beta_{3} + 63 \beta_{4} - 27 \beta_{5} ) q^{27} \) \( + ( 134 + 84 \beta_{1} - 84 \beta_{2} - 30 \beta_{4} - 30 \beta_{5} ) q^{28} \) \( + ( -108 + 127 \beta_{2} - 54 \beta_{3} - 63 \beta_{4} ) q^{29} \) \( + ( -168 - 36 \beta_{1} - 114 \beta_{2} + 144 \beta_{3} - 3 \beta_{4} + 21 \beta_{5} ) q^{30} \) \( + ( 467 - 55 \beta_{1} - 110 \beta_{2} + 421 \beta_{3} - 92 \beta_{4} + 46 \beta_{5} ) q^{31} \) \( + ( -96 - 129 \beta_{1} + 81 \beta_{3} - 15 \beta_{5} ) q^{32} \) \( + ( -338 + 83 \beta_{1} - 35 \beta_{2} - 507 \beta_{3} + 91 \beta_{4} - 53 \beta_{5} ) q^{33} \) \( + ( -6 - 30 \beta_{1} - 15 \beta_{2} - 189 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} ) q^{34} \) \( + ( 315 + 155 \beta_{1} + 155 \beta_{2} + 702 \beta_{3} + 36 \beta_{4} - 36 \beta_{5} ) q^{35} \) \( + ( -672 + 108 \beta_{1} + 255 \beta_{2} - 801 \beta_{3} - 21 \beta_{4} + 84 \beta_{5} ) q^{36} \) \( + ( 128 - 126 \beta_{1} + 126 \beta_{2} + 36 \beta_{4} + 36 \beta_{5} ) q^{37} \) \( + ( -270 - 115 \beta_{2} - 135 \beta_{3} + 27 \beta_{4} ) q^{38} \) \( + ( 683 + 165 \beta_{1} + 152 \beta_{2} + 717 \beta_{3} - 14 \beta_{4} - 10 \beta_{5} ) q^{39} \) \( + ( -536 - 10 \beta_{1} - 20 \beta_{2} - 492 \beta_{3} + 88 \beta_{4} - 44 \beta_{5} ) q^{40} \) \( + ( 891 + 86 \beta_{1} - 783 \beta_{3} + 108 \beta_{5} ) q^{41} \) \( + ( 1940 - 356 \beta_{1} - 226 \beta_{2} + 489 \beta_{3} - 112 \beta_{4} - 55 \beta_{5} ) q^{42} \) \( + ( -170 + 332 \beta_{1} + 166 \beta_{2} + 176 \beta_{3} + 85 \beta_{4} - 170 \beta_{5} ) q^{43} \) \( + ( 660 - 241 \beta_{1} - 241 \beta_{2} + 1026 \beta_{3} - 147 \beta_{4} + 147 \beta_{5} ) q^{44} \) \( + ( -525 - 177 \beta_{1} + 78 \beta_{2} - 18 \beta_{3} - 174 \beta_{4} - 57 \beta_{5} ) q^{45} \) \( + ( -832 - 128 \beta_{1} + 128 \beta_{2} + 29 \beta_{4} + 29 \beta_{5} ) q^{46} \) \( + ( -2430 - 311 \beta_{2} - 1215 \beta_{3} + 120 \beta_{4} ) q^{47} \) \( + ( -532 + 275 \beta_{1} + 50 \beta_{2} + 1431 \beta_{3} - 18 \beta_{4} - 31 \beta_{5} ) q^{48} \) \( + ( -1010 - 151 \beta_{1} - 302 \beta_{2} - 891 \beta_{3} + 238 \beta_{4} - 119 \beta_{5} ) q^{49} \) \( + ( 192 + 439 \beta_{1} - 189 \beta_{3} + 3 \beta_{5} ) q^{50} \) \( + ( 216 + 126 \beta_{1} + 81 \beta_{2} - 783 \beta_{3} - 171 \beta_{4} + 270 \beta_{5} ) q^{51} \) \( + ( 108 - 252 \beta_{1} - 126 \beta_{2} + 8 \beta_{3} - 54 \beta_{4} + 108 \beta_{5} ) q^{52} \) \( + ( 1602 - 198 \beta_{1} - 198 \beta_{2} + 3132 \beta_{3} - 36 \beta_{4} + 36 \beta_{5} ) q^{53} \) \( + ( -1908 + 180 \beta_{1} + 9 \beta_{2} - 3024 \beta_{3} + 252 \beta_{4} - 234 \beta_{5} ) q^{54} \) \( + ( 365 - 29 \beta_{1} + 29 \beta_{2} - 124 \beta_{4} - 124 \beta_{5} ) q^{55} \) \( + ( 432 + 718 \beta_{2} + 216 \beta_{3} - 84 \beta_{4} ) q^{56} \) \( + ( 1236 - 88 \beta_{1} + 81 \beta_{2} + 507 \beta_{3} - 7 \beta_{4} ) q^{57} \) \( + ( 2416 + 8 \beta_{1} + 16 \beta_{2} + 2289 \beta_{3} - 254 \beta_{4} + 127 \beta_{5} ) q^{58} \) \( + ( 933 - 802 \beta_{1} - 972 \beta_{3} - 39 \beta_{5} ) q^{59} \) \( + ( 944 + 238 \beta_{1} + 38 \beta_{2} + 228 \beta_{3} + 416 \beta_{4} - 214 \beta_{5} ) q^{60} \) \( + ( 58 - 166 \beta_{1} - 83 \beta_{2} - 1264 \beta_{3} - 29 \beta_{4} + 58 \beta_{5} ) q^{61} \) \( + ( -1596 + 1000 \beta_{1} + 1000 \beta_{2} - 2862 \beta_{3} + 165 \beta_{4} - 165 \beta_{5} ) q^{62} \) \( + ( 48 + 192 \beta_{1} - 303 \beta_{2} + 2619 \beta_{3} - 36 \beta_{4} + 462 \beta_{5} ) q^{63} \) \( + ( 2074 + 501 \beta_{1} - 501 \beta_{2} + 111 \beta_{4} + 111 \beta_{5} ) q^{64} \) \( + ( -1728 - 287 \beta_{2} - 864 \beta_{3} - 81 \beta_{4} ) q^{65} \) \( + ( -2202 - 987 \beta_{1} - 519 \beta_{2} - 189 \beta_{3} + 153 \beta_{4} + 48 \beta_{5} ) q^{66} \) \( + ( -745 + 554 \beta_{1} + 1108 \beta_{2} - 938 \beta_{3} - 386 \beta_{4} + 193 \beta_{5} ) q^{67} \) \( + ( -1380 - 123 \beta_{1} + 999 \beta_{3} - 381 \beta_{5} ) q^{68} \) \( + ( -2659 + 85 \beta_{1} + 386 \beta_{2} - 780 \beta_{3} - 40 \beta_{4} - 253 \beta_{5} ) q^{69} \) \( + ( 310 + 176 \beta_{1} + 88 \beta_{2} + 3471 \beta_{3} - 155 \beta_{4} + 310 \beta_{5} ) q^{70} \) \( + ( -1050 - 876 \beta_{1} - 876 \beta_{2} - 1188 \beta_{3} + 456 \beta_{4} - 456 \beta_{5} ) q^{71} \) \( + ( 1428 - 897 \beta_{1} - 804 \beta_{2} - 99 \beta_{3} - 66 \beta_{4} - 21 \beta_{5} ) q^{72} \) \( + ( -2734 + 297 \beta_{1} - 297 \beta_{2} - 27 \beta_{4} - 27 \beta_{5} ) q^{73} \) \( + ( 5724 - 610 \beta_{2} + 2862 \beta_{3} - 378 \beta_{4} ) q^{74} \) \( + ( 2744 - 241 \beta_{1} - 436 \beta_{2} - 1611 \beta_{3} - 102 \beta_{4} + 185 \beta_{5} ) q^{75} \) \( + ( -1248 + 43 \beta_{1} + 86 \beta_{2} - 1277 \beta_{3} - 58 \beta_{4} + 29 \beta_{5} ) q^{76} \) \( + ( 9 + 1283 \beta_{1} + 216 \beta_{3} + 225 \beta_{5} ) q^{77} \) \( + ( -100 + 412 \beta_{1} + 668 \beta_{2} + 3108 \beta_{3} - 139 \beta_{4} + 317 \beta_{5} ) q^{78} \) \( + ( -344 - 646 \beta_{1} - 323 \beta_{2} - 2083 \beta_{3} + 172 \beta_{4} - 344 \beta_{5} ) q^{79} \) \( + ( -2184 - 410 \beta_{1} - 410 \beta_{2} - 5076 \beta_{3} - 354 \beta_{4} + 354 \beta_{5} ) q^{80} \) \( + ( 4617 + 837 \beta_{1} + 648 \beta_{2} + 3807 \beta_{3} + 108 \beta_{4} - 405 \beta_{5} ) q^{81} \) \( + ( -1072 - 545 \beta_{1} + 545 \beta_{2} + 86 \beta_{4} + 86 \beta_{5} ) q^{82} \) \( + ( -54 + 1297 \beta_{2} - 27 \beta_{3} + 756 \beta_{4} ) q^{83} \) \( + ( -3318 + 938 \beta_{1} + 1440 \beta_{2} - 4650 \beta_{3} - 16 \beta_{4} - 342 \beta_{5} ) q^{84} \) \( + ( 2958 - 168 \beta_{1} - 336 \beta_{2} + 3456 \beta_{3} + 996 \beta_{4} - 498 \beta_{5} ) q^{85} \) \( + ( -3498 - 1087 \beta_{1} + 3996 \beta_{3} + 498 \beta_{5} ) q^{86} \) \( + ( -4712 - 628 \beta_{1} - 899 \beta_{2} - 2505 \beta_{3} - 296 \beta_{4} + 244 \beta_{5} ) q^{87} \) \( + ( -418 + 310 \beta_{1} + 155 \beta_{2} - 4983 \beta_{3} + 209 \beta_{4} - 418 \beta_{5} ) q^{88} \) \( + ( 3102 + 1470 \beta_{1} + 1470 \beta_{2} + 3996 \beta_{3} - 1104 \beta_{4} + 1104 \beta_{5} ) q^{89} \) \( + ( 3870 + 432 \beta_{1} + 414 \beta_{2} + 594 \beta_{3} - 333 \beta_{4} - 99 \beta_{5} ) q^{90} \) \( + ( 6227 - 421 \beta_{1} + 421 \beta_{2} - 218 \beta_{4} - 218 \beta_{5} ) q^{91} \) \( + ( 6588 - 1042 \beta_{2} + 3294 \beta_{3} + 288 \beta_{4} ) q^{92} \) \( + ( -235 + 1101 \beta_{1} - 64 \beta_{2} - 5736 \beta_{3} - 158 \beta_{4} - 145 \beta_{5} ) q^{93} \) \( + ( -6122 - 1264 \beta_{1} - 2528 \beta_{2} - 5811 \beta_{3} + 622 \beta_{4} - 311 \beta_{5} ) q^{94} \) \( + ( -1626 + 196 \beta_{1} + 1512 \beta_{3} - 114 \beta_{5} ) q^{95} \) \( + ( 522 + 27 \beta_{1} - 909 \beta_{2} + 2862 \beta_{3} - 225 \beta_{4} + 117 \beta_{5} ) q^{96} \) \( + ( 244 - 784 \beta_{1} - 392 \beta_{2} + 9383 \beta_{3} - 122 \beta_{4} + 244 \beta_{5} ) q^{97} \) \( + ( -2910 - 1509 \beta_{1} - 1509 \beta_{2} - 4914 \beta_{3} + 453 \beta_{4} - 453 \beta_{5} ) q^{98} \) \( + ( 984 + 141 \beta_{1} + 1542 \beta_{2} + 3771 \beta_{3} + 165 \beta_{4} + 3 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 15q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut -\mathstrut 99q^{6} \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 99q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 15q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut -\mathstrut 99q^{6} \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 99q^{9} \) \(\mathstrut -\mathstrut 36q^{10} \) \(\mathstrut +\mathstrut 483q^{11} \) \(\mathstrut +\mathstrut 330q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 1146q^{14} \) \(\mathstrut -\mathstrut 1026q^{15} \) \(\mathstrut +\mathstrut 15q^{16} \) \(\mathstrut +\mathstrut 1404q^{18} \) \(\mathstrut -\mathstrut 258q^{19} \) \(\mathstrut +\mathstrut 1614q^{20} \) \(\mathstrut +\mathstrut 480q^{21} \) \(\mathstrut -\mathstrut 369q^{22} \) \(\mathstrut -\mathstrut 282q^{23} \) \(\mathstrut -\mathstrut 1449q^{24} \) \(\mathstrut -\mathstrut 273q^{25} \) \(\mathstrut +\mathstrut 54q^{27} \) \(\mathstrut +\mathstrut 1308q^{28} \) \(\mathstrut -\mathstrut 1056q^{29} \) \(\mathstrut -\mathstrut 1278q^{30} \) \(\mathstrut +\mathstrut 1290q^{31} \) \(\mathstrut -\mathstrut 1161q^{32} \) \(\mathstrut +\mathstrut 279q^{33} \) \(\mathstrut +\mathstrut 513q^{34} \) \(\mathstrut -\mathstrut 2385q^{36} \) \(\mathstrut +\mathstrut 12q^{37} \) \(\mathstrut -\mathstrut 789q^{38} \) \(\mathstrut +\mathstrut 1974q^{39} \) \(\mathstrut -\mathstrut 1314q^{40} \) \(\mathstrut +\mathstrut 7629q^{41} \) \(\mathstrut +\mathstrut 9612q^{42} \) \(\mathstrut -\mathstrut 285q^{43} \) \(\mathstrut -\mathstrut 4212q^{45} \) \(\mathstrut -\mathstrut 5760q^{46} \) \(\mathstrut -\mathstrut 9642q^{47} \) \(\mathstrut -\mathstrut 6771q^{48} \) \(\mathstrut -\mathstrut 1863q^{49} \) \(\mathstrut +\mathstrut 3027q^{50} \) \(\mathstrut +\mathstrut 2457q^{51} \) \(\mathstrut -\mathstrut 240q^{52} \) \(\mathstrut -\mathstrut 405q^{54} \) \(\mathstrut +\mathstrut 2016q^{55} \) \(\mathstrut -\mathstrut 462q^{56} \) \(\mathstrut +\mathstrut 5367q^{57} \) \(\mathstrut +\mathstrut 6462q^{58} \) \(\mathstrut +\mathstrut 6225q^{59} \) \(\mathstrut +\mathstrut 7470q^{60} \) \(\mathstrut +\mathstrut 3630q^{61} \) \(\mathstrut -\mathstrut 7578q^{63} \) \(\mathstrut +\mathstrut 15450q^{64} \) \(\mathstrut -\mathstrut 7158q^{65} \) \(\mathstrut -\mathstrut 13734q^{66} \) \(\mathstrut -\mathstrut 5055q^{67} \) \(\mathstrut -\mathstrut 10503q^{68} \) \(\mathstrut -\mathstrut 13878q^{69} \) \(\mathstrut -\mathstrut 9684q^{70} \) \(\mathstrut +\mathstrut 8451q^{72} \) \(\mathstrut -\mathstrut 14622q^{73} \) \(\mathstrut +\mathstrut 26454q^{74} \) \(\mathstrut +\mathstrut 21021q^{75} \) \(\mathstrut -\mathstrut 4047q^{76} \) \(\mathstrut +\mathstrut 2580q^{77} \) \(\mathstrut -\mathstrut 12060q^{78} \) \(\mathstrut +\mathstrut 4764q^{79} \) \(\mathstrut +\mathstrut 18387q^{81} \) \(\mathstrut -\mathstrut 9702q^{82} \) \(\mathstrut -\mathstrut 1866q^{83} \) \(\mathstrut -\mathstrut 6486q^{84} \) \(\mathstrut +\mathstrut 12366q^{85} \) \(\mathstrut -\mathstrut 37731q^{86} \) \(\mathstrut -\mathstrut 21564q^{87} \) \(\mathstrut +\mathstrut 14787q^{88} \) \(\mathstrut +\mathstrut 20790q^{90} \) \(\mathstrut +\mathstrut 34836q^{91} \) \(\mathstrut +\mathstrut 33636q^{92} \) \(\mathstrut +\mathstrut 19254q^{93} \) \(\mathstrut -\mathstrut 12708q^{94} \) \(\mathstrut -\mathstrut 13362q^{95} \) \(\mathstrut -\mathstrut 3672q^{96} \) \(\mathstrut -\mathstrut 28959q^{97} \) \(\mathstrut -\mathstrut 9126q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(x^{5}\mathstrut +\mathstrut \) \(11\) \(x^{4}\mathstrut +\mathstrut \) \(14\) \(x^{3}\mathstrut +\mathstrut \) \(98\) \(x^{2}\mathstrut +\mathstrut \) \(20\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} + 11 \nu^{4} - 121 \nu^{3} + 98 \nu^{2} + 1118 \nu - 220 \)\()/1098\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 11 \nu^{4} + 121 \nu^{3} - 98 \nu^{2} + 529 \nu + 220 \)\()/549\)
\(\beta_{3}\)\(=\)\((\)\( 55 \nu^{5} - 56 \nu^{4} + 616 \nu^{3} + 649 \nu^{2} + 5488 \nu + 22 \)\()/1098\)
\(\beta_{4}\)\(=\)\((\)\( 373 \nu^{5} - 260 \nu^{4} + 3958 \nu^{3} + 6817 \nu^{2} + 37558 \nu + 15082 \)\()/1098\)
\(\beta_{5}\)\(=\)\((\)\( -406 \nu^{5} + 623 \nu^{4} - 4657 \nu^{3} - 3583 \nu^{2} - 36898 \nu + 6206 \)\()/1098\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(21\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(22\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\) \(\beta_{1}\mathstrut -\mathstrut \) \(26\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(22\) \(\beta_{5}\mathstrut -\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(237\) \(\beta_{3}\mathstrut -\mathstrut \) \(23\) \(\beta_{2}\mathstrut -\mathstrut \) \(46\) \(\beta_{1}\mathstrut +\mathstrut \) \(22\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(23\) \(\beta_{5}\mathstrut -\mathstrut \) \(46\) \(\beta_{4}\mathstrut +\mathstrut \) \(549\) \(\beta_{3}\mathstrut -\mathstrut \) \(270\) \(\beta_{2}\mathstrut -\mathstrut \) \(135\) \(\beta_{1}\mathstrut +\mathstrut \) \(572\)\()/3\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1 + \beta_{3}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
1.89154 3.27625i
−0.102534 + 0.177594i
−1.28901 + 2.23263i
1.89154 + 3.27625i
−0.102534 0.177594i
−1.28901 2.23263i
−5.67463 3.27625i −1.11837 8.93024i 13.4676 + 23.3266i 10.2044 5.89150i −22.9114 + 54.3399i 26.6364 46.1356i 71.6534i −78.4985 + 19.9746i −77.2081
2.2 0.307601 + 0.177594i 8.32172 + 3.42768i −7.93692 13.7472i −30.0804 + 17.3669i 1.95104 + 2.53225i 15.6054 27.0294i 11.3212i 57.5020 + 57.0484i −12.3370
2.3 3.86703 + 2.23263i −8.70335 2.29167i 1.96929 + 3.41090i 13.8760 8.01130i −28.5397 28.2933i −36.2418 + 62.7727i 53.8574i 70.4965 + 39.8904i 71.5451
5.1 −5.67463 + 3.27625i −1.11837 + 8.93024i 13.4676 23.3266i 10.2044 + 5.89150i −22.9114 54.3399i 26.6364 + 46.1356i 71.6534i −78.4985 19.9746i −77.2081
5.2 0.307601 0.177594i 8.32172 3.42768i −7.93692 + 13.7472i −30.0804 17.3669i 1.95104 2.53225i 15.6054 + 27.0294i 11.3212i 57.5020 57.0484i −12.3370
5.3 3.86703 2.23263i −8.70335 + 2.29167i 1.96929 3.41090i 13.8760 + 8.01130i −28.5397 + 28.2933i −36.2418 62.7727i 53.8574i 70.4965 39.8904i 71.5451
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.3
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(9, [\chi])\).